# Full text of "History Of The Theory Of Numbers - I"

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```60                    HISTORY OF THE THEORY OF NUMBERS.              [CHAP, m
proved in 1681 (p. 50). Mahnke gave reasons (pp. 54-7) for believing that Leibniz rediscovered independently Fermat's theorem before he became acquainted, about 1681-2, with Fermat's Varia opera math, of 1679. In 1682 (p. 42), Leibniz stated that (p— 2)1=1 (mod p) if p is a prime [equivalent to Wilson's theorem], but that (p— 2)l=m (mod p), if p is composite, m having a factor > 1 in common with p.
De la Hire8 stated that if fc2r+1 is divided by 2(2r+l) we get A; as a remainder, perhaps after adding a multiple of the divisor. For example, if Jf is divided by 10 we get the remainder k. He remarked that Can4 had observed that the cube of any number fc<6 has the remainder k when
divided by 6.
L. Euler9 stated Fermat's theorem in the form: If n+l is a prime dividing neither a nor b, then a71— bn is divisible by n+l. He was not able to give a proof at that time. He stated the generalization: If e=pm~1(p — 1) and if p is a prime, the remainder obtained on dividing ae by pm is 0 or 1 [a special case of Euler14]. He stated also that if m, n, p, . . . are distinct primes not dividing a and if A is the L c. m. of m — 1, n — 1, p — 1, . . ., then aA— 1 is divisible by mnp. . . [and ak — 1 by mr n* . . .if k = A mr~ln8~l . . .].
Euler10 first published a proof of Fermat's theorem.   For a prime p,
Hence if ap— a is divisible by p, also (l+a)p— (1+d) is, and hence also (a-f2)p-(a+2),. . ., (a-l-b)p-(a-f 6). Fora=2, 2P- 2 was proved divisible by p. Hence, writing x for 2-j-b, we conclude that zp— x is divisible by p for any integer x.
G. W. Kraft11 proved similarly that 2p-2=wp.
L. Euler' s12 second proof is based, like his first, on the binomial theorem. If a, fe are integers and p is a prime, (a+b)p— ap-fcp is divisible by p. Then, if ap— a and 6P— & are divisible by p, also (a+6)p — a — b is divisible by p. Take 6=1. Thus (a-fl)p— a — 1 is divisible by p if ap — a is. Taking a =1,2, 3,. . . in turn, we conclude that 2P— 2, 3P— 3, . . . , cp— c are divisible by p. -
L. Euler13 preferred his third proof to his earlier proofs since it avoids the use of the binomial theorem. If p is a prune and a is any integer not
*Hist. Acad. Sc. Paris, amide 1704, pp. 42-4; m£m., 358-362.
9Comm. Ac. Petrop., 6, 1732-3, 106; Comm. Arith., 1, 1849, p. 2.    [Opera postuma, I, 1862,